Exercise 9 – Measurements of time and frequency. 1.

Aim of the exercise

The aim of the exercise is to familiarize students with methods of measuring time parameters of electrical signals such as frequency, period and phase shift between the two signals. Additional error evaluation will be described. 2.

Main topics

 Measurements of time parameters (time, frequency, and duty cycle) of electrical signals by using the oscilloscope and Digital Frequency Counter (DFC)  Measurements of the phase shift between the two sinusoidal signals with a frequency counter and a digital oscilloscope. 3.

Gained skills

 the ability to connect the circuit according to the measurement diagram  the ability to measure correctly time parameters (period, frequency, duty cycle) and phase shift using a digital frequency counter and an oscilloscope.  the ability to properly document the experiment results  the ability to determine of measurement errors

4.

Quantities, definition and units

 Time (t): fundamental physical quantity "The second is the duration of 9191631770 periods of the radiation corresponding to the transition between the hyperfine levels of the ground state of the ceasium-133”  Frequency (f): "Frequency (f) O of a periodic phenomenon is the number (n) of occurrences of phenomenon in a time unit (τ); frequency is the reciprocal of the time (T) between successive occurrences of the phenomenon.” “"One hertz is the frequency of a period phenomenon of which the periodic time is one second; one hertz is one cycle per second”.  Pulsatance, ω: "Angular frequency (Pulsatance) is the frequency of a periodic process expressed in radians per second; pulsatance is the rate of change of angle in time.”

5.

Methods of measurements used in the experiment.

There are many methods for measurements of frequency or time. In our experiment only a few of them are used: analog methods based on measurement of time with the oscilloscope, and direct method based on of measurement frequency and time with the multifunction digital counter. Oscilloscope method used in the experiments are extremely simple - they implement either internal (linear) time base, or external reference signal (Lissajous method). Having known the time base speed time/div (a value which may be read from oscilloscope’s screen), all we need to do is to measure the length of one or more cycles of the observed signal.. This method is fast but not very precise. Sinusoidal (external) time-base method (The Lissajous method). An other analog frequency measurement method involving the oscilloscope rely on using the oscilloscope as a kind of null indicator for comparison of a sine signal with unknown frequency with a reference sine signal whose frequency should be well defined and easily varied. One of the signals is fed to the Y channel of the oscilloscope, the second one to the X channel. An interaction of these two signals produces on the display more or less complicated snaky loops, whose form allows for determining the unknown frequency. A typical Lissajous figure is shown in Fig. 1.

Fig. 1 Frequency measurement with the Lissajous method a) measurement diagram, b) Lissajous figure and frequency determination. If we assume that fy is the unknown frequency signal connected to the Y channel and fx is the reference signal connected to the X channel then we have 𝑓𝑦 = 𝑓𝑥

𝑛𝑥 1 𝑑𝜑 + 𝑛𝑦 2𝜋 𝑑𝑡

where nx, ny denote number of intersections of the Lissajous curve with horizontal and vertical axes, respectively, and dφ/dt is the rate of phase change (the trace rotation speed). The reference signal should be adjusted until the displayed figure is possibly stable (dφ/dt ≈ 0). It

is sometimes attainable with difficulty, and needs both signal sources involved having adequate frequency stability. The method's error is roughly equal to relative calibration error of the reference source. Due to the mentioned stability problems, usefulness of the method is limited to rather low frequency applications. Digital counter method. Digital counter methods rely on continuing number of events (in this case – the number of cycles) with the counter open during precisely determined window. The periodic input signal of any shape, including sine waveform, is formed in an input shaper block to have standard form of possibly short pulses that are fed to the counter controlled by an accurate and stable quartz oscillator (see Fig.2).

fx

Input shape r

AND gate

Counter

Display

τ

Fig. 2. Principle of direct digital measurement of frequency If the counter is opened for e.g. 1 second, then the number of pulses counted during this time directly gives the measured frequency. If we denote open gate time as τ, input signal period as Tx, and number of cycles counted as N, then 𝜏 = 𝑁 ∙ 𝑇𝑥 and the unknown frequency is equal to 𝑓𝑥 = 𝑁/𝜏 It is intuitively obvious that the accuracy of this method mainly depends on the accuracy of gate timing. It may be shown that the limiting error of direct frequency measurement method is equal to 𝛿𝑔 𝑓𝑥 = 𝛿𝑔 𝑓𝑟𝑒𝑓 +

1 1 = 𝛿𝑔 𝑓𝑟𝑒𝑓 + 𝑁 𝑓𝑥 ∙ 𝜏

where δgfref is the limiting error of quartz oscillator frequency.5gfk is the limiting.

In conclusion, one may see that the limiting error of direct frequency measurement method decreases - as the number N increases, i.e. measured frequency rises, - gating time τ is longer. The accuracy of the method is getting worse if measured frequency goes down. The way out of this problem is to switch to digitally measure the period of the signal T x instead of directly measure the frequency. In this method, the gate is opened for the time of one period of the input signal, and during this time pulses with the reference frequency f w produced by quartz oscillator are counted. The reference frequency fref may be changed by adjusting the frequency dividing factor m, so that fw = fref/m. Finally, the frequency we are looking for is equal to f x = 1/Tx. The limiting error of measured period is equal to 𝛿𝑔 𝑇𝑥 = 𝛿𝑔 𝑓𝑟𝑒𝑓 +

1 𝑚𝑓𝑥 = 𝛿𝑔 𝑓𝑟𝑒𝑓 + 𝑁 𝑓𝑟𝑒𝑓

and 𝛿𝑔 𝑓𝑥 = 𝛿𝑔 𝑇𝑥 The limiting error of this method of frequency measurement may be further decreased if we measure not the one period of input signal but time of few periods, say k periods. In this case, the limiting error is expressed as 𝛿𝑔 𝑓𝑥 = 𝛿𝑔 𝑓𝑟𝑒𝑓 +

1 𝑚𝑓𝑥 = 𝛿𝑔 𝑓𝑟𝑒𝑓 + 𝑘𝑁 𝑓𝑟𝑒𝑓 ∙ 𝑘

The universal frequency/time counter may also be used for measurement of time interval. In this case the first event opens the gate and the second one stops it. In between the counter count pulses with reference frequency coming from quartz oscillator. It is worth knowing that modem time-frequency counters are capable to select appropriate mode of operation automatically, thus minimizing the quantization error. Moreover, the timer-counter Keysight 53220A measure only the period of input signal. So, measured frequency is always calculated. A part of the limiting error of discussed measurements comes from the finite time of gate switching. The detailed discussion of that problem is beyond the scope of this text.

6.

Instrumentation

Laboratory module description

Fig. 1. Module F01. G1 – sinusoidal waveform output, G2 – rectangular waveform output, PF – phase shifter 7.

Measurements and tasks

Table 1. Measurement functions of 53220A Keysight digital frequency counter Function Period Frequency Pulse counting Duty cycle Pulse width Pulse distance Phase shift

Symbol T f n   t 

Settings FREQ/PERIOD  PERIOD FREQ/PERIOD  FREQ TOTALIZE  GATED TIME INTERVAL  DUTY CYCLE TIME INTERVAL  PULSE WIDTH TIME INTERVAL  PULSE WIDTH TIME INTERVAL  PHASE  PHASE 1-2

Table 2. Measurement functions of 1052 Rigol oscilloscope Function Period Frequency Duty cycle Pulse width Pulse distance Phase shift

Symbol T f   t 

Settings MEASURE  TIME  PERIOD MEASURE  TIME  FREQ MEASURE  TIME  +DUTY MEASURE  TIME  +WIDTH MEASURE  TIME  -WIDTH MEASURE  TIME  PHAS 12

Note: Before the measurements: - connect power supply to the F01 Module. - set a frequency counter impedance to 50 Ω for inputs 1 and 2: (1 or 2)  Impedance: 50  Ω. Task 1. Measurements of time parameters of sinusoidal signals

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1.1 Use the DFC 53220A (Digital Frequency Counter) to measure time parameters (period and frequency) of sine signal from generator G1 output in F01 module. NOTE: Prior to the measurements assess the stability of the G1 generator and adjust the resolution of the frequency counter. To do this, set the number of digits displayed on the frequency counter display in a way to achieve observed instability of the result on the least significant digit ( Digits  AutoDigits: Off, and then use the knob to set number of digits). To make reading the results easier, the manual trigger of measurements can be used. Once you activate following sequence of commands:  Trigger  Source  Manual, each time you press the button Trigger, a single measurement will be performed. To go back to automatic trigger mode execution following sequence of commands:  Trigger  Source  Internal. Measure appropriate parameters using following functions:

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a) PERIOD – measure the period of the signal with gating time of 1 s. If selected gating time is longer than measured period, the DFC measures an average period.

 

b) FREQ – measure the frequency of the signal with gating time of 1 s. In this case, the frequency is measured with the indirect method.

!

Compare and comment obtained results from PERIOD, FREQ and TOTALIZE:GATED measurement method.

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1.2 Use the oscilloscope to measure time parameters (period and frequency) of sine signal from generator G1 output in F01 module.

c) TOTALIZE:GATED – counting the number of pulses during 1 s of Gate Time. This case realizes the direct frequency measurement method (using frequency definition – counting of phenomena occurrences during reference time interval)

Take measurements of the period and frequency with the use the manual procedure (length measurements) and the automatic measurement functions. Set the oscilloscope to minimize measurements errors.

 !

Insert the oscillograms into the report. Write down Cx constant and measurement results. Estimate the limiting error of both measurements. Compare and comment obtained results of period and frequency measurements with the corresponding values obtained in point 1.1a and 1.1b. Assume the DFC as a reference instrument.

Task 2. Measurements of time parameters of rectangular signals.

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2.1 Use the DFC 53220A (Digital Frequency Counter) to measure time parameters (period and frequency) of sine signal from generator G2 output in F01 module. Prior to the measurements assess the stability of the G2 generator and adjust the resolution of the frequency counter. To do this, set the number of digits displayed on the frequency counter display in a way to achieve observed instability of the result on the least significant digit.

 !

   !

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a) Measure the period and frequency using the PERIOD and FREQ functions respectively. Set the gating time of 1 s. Compare and comment on the results. b) Measure the pulse width of rectangular waveform (Width: Pos) and time interval between pulses (Width: Neg). c) Determine duty cycle of the signal based on the parameters obtained in point a) and b). d) Measure the duty cycle of the signal using automatic function in the DFC (DutyCycle: Pos). Compare and comment results obtained in points c) and d). 2.2 Use the oscilloscope to measure time parameters (period and frequency) of sine signal from generator G2 output in F01 module

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a) Take measurements of the period and frequency with the use the manual procedure (length measurements) and the automatic measurement functions. Set the oscilloscope to minimize measurements errors.

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Insert the oscillograms into the report. Write down Cx constant and measurement results. Estimate the limiting error of both measurements.

!

Compare obtained results of period and frequency measurements with the corresponding values obtained in point 2.1a. Assume the DFC as a reference instrument.

  

b) Take measurements of the pulse width and time interval with the use the manual procedure and the automatic measurement functions (WIDTH+ and WIDTH-). Insert the oscillograms into the report. Estimate the limiting error of the measurements. c) Based on results in point a) and b), determine value of the duty cycle and estimate the limiting error of both results.

 d) Take measurements of the duty cycle with the use of automatic function +DUTY. !

Compare values of duty cycle obtained in points 2.1c and 2.1d. Assume the DFC as a reference instrument Task 3. Measurements of phase shift of sinusoidal signals.

Connect the output of the G1 generator to the input of the phase shifter PF in F01 module.  3.1 Use the DFC to measure the phase shift between signals on the input and output of the PF module. To gain measurement result in range of 0 - 360ºset option:  Time Interval  Phase  Phase Meas: 0–360. Use the oscilloscope to measure phase shift between signals on the input and output of the  3.2 PF module. Use the sine signal from G1 as an input signal for the phase shifter (PS). measurements with the use the manual procedure (length measurements. Set the  Take oscilloscope to minimize measurements errors. Insert the oscillogram into the report and mark all sectors used for ε calculation.

 Estimate the limiting error of phase shift measurement. !

Compare and comment obtained result of phase shift measurement with the corresponding value of phase shift obtained in point 3.1.

8.

Control questions 1. Describe how one can measure frequency with the use of digital oscilloscope. 2. Describe how one can measure phase shift between two signals with the use of dual channel oscilloscope. 3. Is it possible to measure the phase shift between two signals of the same shape, amplitude and frequency but from two independent sources? Justify your answer. 4. What is the role of the internal generator in the frequency counter? 5. For 1 ms/div time scale a waveform as in figure below has been observed at the oscilloscope:

Estimate the signal frequency and the limiting error (absolute and relative) knowing that the formula for the relative limiting error is expressed by formula: 𝛿𝑔 𝑓𝑥 = 3% +

0,1 𝑑𝑖𝑣 ∙ 100% 𝑙𝑙𝑥

Where lx is the measured length. 6. At what timescale (1 µs or 2µs) the oscilloscope offers a smaller relative error in measurement of the frequency of the signal of 1/6 MHz with the use of the method of the length measurement? The oscilloscope screen has 10 divs in horizontal direction and the formula for the error is: 𝛿𝑔 𝑇𝑥 = 3% +

0,1 𝑑𝑖𝑣 ∙ 100% 𝑙𝑙𝑥

7. For 1 ms/div time scale a waveform as in figure below has been observed at the oscilloscope:

Estimate the duty cycle and the limiting error of this measurement knowing that the limiting error formula is: 𝛿𝑔 𝜏𝑥 = 50𝑝𝑝𝑚 +

0,1 𝑑𝑖𝑣 ∙ 100% 𝑙𝑙𝑥

8. Derive the formula for the limiting error of the duty cycle of the rectangular signal measurement with the method of the length measurements with the digital oscilloscope. The parameter is defined as follows: a) T b) t)

9. Derive the formula for the limiting error of the phase shift measurement with the method of the length measurement with the digital oscilloscope. 10. Describe the direct method of the frequency measurement with the digital frequency counter and propose the analysis of the measurement error. 11. Describe the indirect method of the frequency measurement with the digital frequency counter and propose the analysis of the measurement error. Explain the difference between measuring period and averaged period. 12. Neglecting the instability of the quartz generator estimate the value of the measured frequency fx for which the measurement error of the direct measurement gets smaller than the measurement error of the indirect measurement (period measurement). 13. For which frequency range (high of low) one should apply indirect frequency measurement method and for which a direct method? 14. Digital frequency counter measured 2500 pulses for the gate of 2 s. Calculate the period of the signal and the limiting error of the measurement. Neglect the quartz oscillator instability. 15. Digital frequency counter measured 5000 pulses for the gate of 1 s and 50002 pulses for 10 s. Which measurement is more accurate? Justify your answer. 16. How long should the direct frequency measurement take if the 10 kHz signal measured with the digital frequency counter in a way that the error component due to the instability of the internal quartz generator of 10-7 constituted a half of the total limiting error?